Lie Representations Research Articles

Mathcal{N} = 4 SYM, (super)-polynomial rings and emergent quantum mechanical symmetries

The structure of half-BPS representations of psu(2, 2|4) leads to the definition of a super-polynomial ring mathcal{R} (8|8) which admits a realisation of psu(2, 2|4) in terms of differential operators on the super-ring. The character of the half-BPS fundamental field representation encodes the resolution of the representation in terms of an exact sequence of modules of mathcal{R} (8|8). The half-BPS representation is realized by quotienting the super-ring by a quadratic ideal, equivalently by setting to zero certain quadratic polynomials in the generators of the super-ring. This description of the half-BPS fundamental field irreducible representation of psu(2, 2|4) in terms of a super-polynomial ring is an example of a more general construction of lowest-weight representations of Lie (super-) algebras using polynomial rings generated by a commuting subspace of the standard raising operators, corresponding to positive roots of the Lie (super-) algebra. We illustrate the construction using simple examples of representations of su(3) and su(4). These results lead to the definition of a notion of quantum mechanical emergence for oscillator realisations of symmetries, which is based on ideals in the ring of polynomials in the creation operators.

Coverings and nonlocal symmetries as well as fundamental solutions of nonlinear equations derived from the nonisospectral AKNS hierarchy

In the paper, we apply a new scheme to generate nonisospectral integrable hierarchies of evolution equations. First of all, we work out the positive-order and the negative-order nonisospectral AKNS hierarchies. Next we introduce a kind of loop algebra from which an expanding nonisospectral integrable model is derived. Specially, the expanding integrable model reduces to the well-known bond pricing equation and the interest rate modeling. By using the Lie group theory we investigate the Lie representations and characteristic solutions of a generalized bond pricing equation from the positive-order nonisospectral integrable equation, and with the help of the characteristic solutions and the Laplace transformation, we again obtain the fundamental solutions of the generalized bond pricing equation. Besides, we apply the Lie group analysis to deduce the invariant solutions and the nonlinear self-adjointness as well as conservation laws, non-invariant solutions of the generalized bond pricing equation. Some reductions of the negative-order nonisospectral integrable hierarchy are also produced. Two of them are the Sine-Gordan equation and the Sinh-Gordan equation. Finally, we investigate the coverings and the nonlocal symmetries of the generalized bond pricing equation and a generalized KdV equation, respectively, although the bond equation is linear which has many applications in the aspect of finance field, by applying the classical Frobenius theorem and the coordinates of a infinitely-dimensional manifold in the form of Cartesian product.

The plethystic inverse of the odd Lie representations

The Frobenius characteristic of L i e n Lie_n , the representation of the symmetric group S n S_n afforded by the multilinear part of the free Lie algebra, is known to satisfy many interesting plethystic identities. In this paper we prove a conjecture of Richard Stanley establishing the plethystic inverse of the sum ∑ n ≥ 0 L i e 2 n + 1 \sum _{n\geq 0} Lie_{2n+1} of the odd Lie characteristics. We obtain an apparently new plethystic decomposition of the regular representation of S n S_n in terms of irreducibles indexed by hooks, and the Lie representations. We also determine the plethystic inverse of the alternating sum of the odd Lie characteristics.

ACTIONS OF MONOIDAL CATEGORIES AND REPRESENTATIONS OF CARTAN TYPE LIE ALGEBRAS

AbstractUsing crossed homomorphisms, we show that the category of weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs) is a left module category over the monoidal category of representations of Lie algebras. In particular, the corresponding bifunctor of monoidal categories is established to give new weak representations (respectively admissible representations) of Lie–Rinehart algebras (respectively Leibniz pairs). This generalises and unifies various existing constructions of representations of many Lie algebras by using this new bifunctor. We construct some crossed homomorphisms in different situations and use our actions of monoidal categories to recover some known constructions of representations of various Lie algebras and to obtain new representations for generalised Witt algebras and their Lie subalgebras. The cohomology theory of crossed homomorphisms between Lie algebras is introduced and used to study linear deformations of crossed homomorphisms.

Prime power variations of higher Lien modules

We define, for each subset S of the set P of primes, an Sn-module LienS with interesting properties. Lien∅ is the well-known representation Lien of Sn afforded by the free Lie algebra, while LienP is the module Conjn of the conjugacy action of Sn on n-cycles. For arbitrary S the module LienS interpolates between the representations Lien and Conjn. We consider the symmetric and exterior powers of LienS. These are the analogues of the higher Lie modules of Thrall. We show that the Frobenius characteristic of these higher LienS modules can be elegantly expressed as a multiplicity-free sum of power sums. In particular this establishes the Schur positivity of new classes of sums of power sums.More generally, for each nonempty subset T of positive integers we define a sequence of symmetric functions fnT of homogeneous degree n. We show that the series ∑λ,λi∈Tpλ can be expressed as symmetrised powers of the functions fnT, analogous to the higher Lie modules first defined by Thrall. This in turn allows us to unify previous results on the Schur positivity of multiplicity-free sums of power sums, as well as investigate new ones. We also uncover some curious plethystic relationships between fnT, the conjugacy action and the Lie representation.

GroupMath: A Mathematica package for group theory calculations

GroupMath is a Mathematica package which performs several calculations related to semi-simple Lie algebras and the permutation groups, both of which are important in various areas of physics. Having in mind the specific needs of theoretical particle physicists, the program computes several basis-independent quantities (such as roots, weights, decomposition of products of representations, branching rules) as well as basis-dependent ones (including explicit representation matrices and Clebsch-Gordon coefficients). Program summaryProgram Title: GroupMath: A Mathematica package for group theory calculationsCPC Library link to program files:https://doi.org/10.17632/hdkksr6v7t.1Developer's repository link:renatofonseca.net/groupmathLicensing provisions: GNU General Public License 3Programming language: MathematicaNature of problem: Computations involving semi-simple Lie algebras are commonplace in several areas of research, such as particle physics. Indeed model builders trying to extend the Standard Model often need to consider new fields which transform as irreducible representations of these algebras. It is therefore convenient to have a computer program which can perform these calculations systematically. Such code might be used directly by the user, or be incorporated in a longer chain of programs. The permutation groups Sm are also important, both by themselves and also as a tool to better describe the properties of the representations of semi-simple Lie algebras.Solution method: GroupMath performs several computations related to semi-simple Lie algebras which are important in high energy physics: Cartan matrices, roots, weights, Weyl reflections, decomposition of products of representations, subgroups, branching rules, and others. Algorithms available in the literature are used to efficiently perform some of these calculations. The program also computes basis-dependent quantities such as representation matrices and Clebsch-Gordon coefficients. Several elements of the theory of finite permutation groups have also been implemented, including Young diagrams and tableaux, the decomposition of products of representations, branching rules, and explicit representation matrices.Additional comments including restrictions and unusual features: Most of GroupMath's functions can handle arbitrary representations of any semi-simple Lie algebra and any permutation group. However, memory and time constraints are important when considering large and/or numerous representations. This is particularly true for the functions which perform basis-dependent computations.

Strict algebraic models for rational parametrised spectra, I

Building on Quillen's rational homotopy theory, we obtain algebraic models for the rational homotopy theory of parametrised spectra. For any simply-connected space $X$ there is a dg Lie algebra $\Lambda_X$ and a (coassociative cocommutative) dg coalgebra $C_X$ that model the rational homotopy type. In this article, we prove that the rational homotopy type of an $X$-parametrised spectrum is completely encoded by a $\Lambda_X$-representation and also by a $C_X$-comodule. The correspondence between rational parametrised spectra and algebraic data is obtained by means of symmetric monoidal equivalences of homotopy categories that vary pseudofunctorially in the simply-connected parameter space $X$. Our results establish a comprehensive dictionary enabling the translation of topological constructions into homological algebra using Lie representations and comodules, and conversely. For example, the fibrewise smash product of parametrised spectra is encoded by the derived tensor product of dg Lie representations and also by the derived cotensor product of dg comodules. As an application, we obtain algebraic descriptions of rational homotopy classes of fibrewise stable maps, providing new tools for the study of spaces of sections.

Inner symmetries of the spatially singular part of the solutions of the Burgers equation and their Lie representations

We describe two new discrete symmetries of the inviscid Burgers (or Riemann–Hopf) equation ut+uux=0. We derived both of them using a local, formal approach of Hopf algebraic renormalization, a tool recently used in algorithmic computations. We prove that one of them is a Lie point transformation. Symmetries generate new exact solutions from the known solutions and provide useful frames of reference in the study of shock wave formation.

A type B analog of the Lie representation

We describe a type B analog of the much studied Lie representation of the symmetric group. The nth Lie representation of Sn restricts to the regular representation of Sn−1, and our generalization mimics this property. Specifically, we construct a representation of the type B Weyl group Bn that restricts to the regular representation of Bn−1. We view both of these representations as coming from the internal zonotopal algebra of the Gale dual of the corresponding reflection arrangements.

Infinitesimal Automorphisms of VB-Groupoids and Algebroids

Abstract VB-groupoids and algebroids are vector bundle objects in the categories of Lie groupoids and Lie algebroids, respectively, and they are related via the Lie functor. VB-groupoids and algebroids play a prominent role in Poisson and related geometries. Additionally, they can be seen as models for vector bundles over singular spaces. In this paper we study their infinitesimal automorphisms, i.e. vector fields on them generating a flow by diffeomorphisms preserving both the linear and the groupoid/algebroid structures. For a special class of VB-groupoids/algebroids coming from representations of Lie groupoids/algebroids, we prove that infinitesimal automorphisms are the same as multiplicative sections of a certain derivation VB-groupoid/algebroid.

Elementary properties of non-Linear Rossby-Haurwitz planetary waves revisited in terms of the underlying spherical symmetry

We revisit Rossby-Haurwitz planetary wave modes of a two-dimensional fluid along the surface of a rotating planet, as elements of irreducible representations of the so(3) Lie algebra. Key questions addressed are, firstly, why it is that the non-linear self-interaction of any Rossby-Haurwitz wave mode is zero, and secondly, why the phase velocity of these wave modes is insensitive to their orientation with respect to the axis of rotation of the planet, while at the same time the very rotation of the planet is a precondition for the existence of the waves. As we show, answers to both questions can be rooted in Lie group and representation theory. In our study the Rossby-Haurwitz modes emerge in a coordinate-free, as well as in a Ricci tensor rank-free manner. We find them with respect to a continuum of spherical coordinate systems, that are arbitrarily oriented with respect to the planet. Furthermore, we show that, in the same sense in which the Lie derivative of Ricci tensor fields is rankfree, the wave equation for Rossby-Haurwitz modes is rank-free. We find that, for each irreducible representation of so(3), there is a corresponding sufficient condition for existence of Rossby-Haurwitz modes as solutions that are separable with respect to space and time. This condition comes in the form of a system of equations of motion for the coordinate systems. Coordinate systems that move along with Rossby-Haurwitz modes emerge as special cases of these. In these coordinate systems the waves appear as stationary spatial fields, so that the motion of the coordinate system coincides with the wave phase propagation. The general solution of the existence condition is a continuum of moving spherical coordinate systems that precess about the axes of the Rossby-Haurwitz modes. Within a single irreducible representation of so(3), the waves are dispersionless.

Double Lie algebroids and representations up to homotopy

We show that a double Lie algebroid, together with a chosen decomposition, is equivalent to a pair of 2-term representations up to homotopy satisfying compatibility conditions which extend the notion of matched pair of Lie algebroids. We discuss in detail the double Lie algebroids arising from the tangent bundle of a Lie algebroid and the cotangent bundle of a Lie bialgebroid.

Chern–Simons Forms and Higher Character Maps of Lie Representations

This paper is a sequel to our earlier work [BFPRW], where we study the derived representation scheme DRep_{g}(A) parametrizing the representations of a Lie algebra A in a finite-dimensional reductive Lie algebra g. In [BFPRW], we defined two canonical maps Tr_{g}(A): HC^{(r)}(A) \to \H[\DRep_{g}(A)]^G and \Phi_{g}(A): H[\DRep_{g}(A)]^G \to H[\DRep_{h}(A)]^W called the Drinfeld trace and the derived Harish-Chandra homomorphism, respectively. In this paper, we give an explicit formula for the Drinfeld trace in terms of Chern-Simons classes of a canonical g-torsor associated to the pair (A, g). Our construction is inspired by (and, in a sense, dual to) the classical construction of `additive regulator maps' due to Beilinson and Feigin. As a consequence, we show that, if A is an abelian Lie algebra, the composite map Phi_{g}(A) Tr_{g}(A) is represented by a canonical differential operator acting on differential forms on Sym(A) and depending only on the Cartan data (h, W, P), where P is a W-invariant polynomial on h. We derive a combinatorial formula for this operator that plays an important role in the study of derived commuting schemes in [BFPRW].

Representations of Homotopy Lie–Rinehart Algebras

AbstractI propose a definition of left/right connection along a strong homotopy Lie–Rinehart algebra. This allows me to generalise simultaneously representations up to homotopy of Lie algebroids and actions ofL∞algebras on graded manifolds. I also discuss the Schouten-Nijenhuis calculus associated to strong homotopy Lie–Rinehart connections.

A modular version of Klyachko's theorem on Lie representations of the general linear group

AbstractKlyachko, in 1974, considered the tensor and Lie powers of the natural module for the general linear group over a field of characteristic 0 and showed that nearly all of the irreducible submodules of the rth tensor power also occur up to isomorphism as submodules of the rth Lie power. Here we prove an analogue for infinite fields of prime characteristic by showing, with some restrictions on r, that nearly all of the indecomposable direct summands of the rth tensor power also occur up to isomorphism as summands of the rth Lie power.

Mean field theory for U(n) dynamical groups

Algebraic mean field theory (AMFT) is a many-body physics modeling tool which firstly, is a generalization of Hartree–Fock mean field theory, and secondly, an application of the orbit method from Lie representation theory. The AMFT ansatz is that the physical system enjoys a dynamical group, which may be either a strong or a weak dynamical Lie group G. When G is a strong dynamical group, the quantum states are, by definition, vectors in one irreducible unitary representation (irrep) space, and AMFT is equivalent to the Kirillov orbit method for deducing properties of a representation from a direct geometrical analysis of the associated integral co-adjoint orbit. AMFT can be the only tractable method for analyzing some complex many-body systems when the dimension of the irrep space of the strong dynamical group is very large or infinite. When G is a weak dynamical group, the quantum states are not vectors in one irrep space, but AMFT applies if the densities of the states lie on one non-integral co-adjoint orbit. The computational simplicity of AMFT is the same for both strong and weak dynamical groups. This paper formulates AMFT explicitly for unitary Lie algebras, and applies the general method to the Lipkin–Meshkov–Glick (2) model and the Elliott (3) model. When the energy in the (3) theory is a rotational scalar function, Marsden–Weinstein reduction simplifies AMFT dynamics to a two-dimensional phase space.

Universally defined representations of Lie conformal superalgebras

We distinguish a class of irreducible finite representations of the conformal Lie (super)algebras. These representations (called universally defined) are the simplest ones from the computational point of view: a universally defined representation of a conformal Lie (super)algebra L is completely determined by commutation relations of L and by the requirement of associative locality of generators. We describe such representations for conformal superalgebras W n , n ≥ 0 , with respect to a natural set of generators. We also consider the problem for superalgebras K n . In particular, we find a universally defined representation for the Neveu–Schwartz conformal superalgebra K 1 and show that the analogues of this representation for n ≥ 2 are not universally defined.

The rank filtration and Robinson’s complex

For a functor from the category of finite sets to abelian groups, Robinson constructed a bicomplex in [A. Robinson, Gamma homology, Lie representations and E ∞ multiplications, Invent. Math. 152 (2) (2003) 331–348] which computes the stable derived invariants of the functor as defined by Dold–Puppe in [A. Dold, D. Puppe, Homologie nicht-additiver Funktoren. Anwendungen., Ann. Inst. Fourier (Grenoble) 11 (1961) 201–312]. We identify a subcomplex of Robinson’s bicomplex which is analogous to a normalization and also computes these invariants. We show that this new bicomplex arises from a natural filtration of the functor obtained by taking left Kan approximations on subcategories of bounded cardinality.

Thermodynamics of finite strain elastic-inelastic deformation

The kinematic relations describing elastic-inelastic deformation that coincide in shape with the well-known Lie representation but are free from the drawback of the latter are extended to the case of thermo-elastic-inelastic deformation with finite strains. The limitations imposed on the kinematics by the principle of objectivity are considered. Relations for the stresses and entropy are derived from the laws of thermodynamics, and a heat-conduction equation is constructed.

A combinatorial proof of Klyachko’s Theorem on Lie representations

Let L be a free Lie algebra of finite rank r over an arbitrary field K of characteristic 0, and let L n denote the homogeneous component of degree n in L. Viewed as a module for the general linear group GL(r,K), L n is known to be semisimple with the isomorphism types of the simple summands indexed by partitions of n with at most r parts. Klyachko proved in 1974 that, for n > 6, almost all such partitions are needed here, the exceptions being the partition with just one part, and the partition in which all parts are equal to 1. This paper presents a combinatorial proof based on the Littlewood-Richardson rule. This proof also yields that if the composition multiplicity of a simple summand in L n is greater than 1, then it is at least \(\frac{n}{6}-1\).